dt/dx= ((x^2+a^2)^(1/2))/v1+((b^2+d^2)^(1/2))/v1
the function dt/dx has a zero at a unique x on (0,d). Use this to justify your discovery that
d= (x^2+a^2)^(1/2) + (b^2+d^2)^(1/2)
as it stands, I think there's something wrong.
you want dt/dx = 0
but since √n is always >= 0, that means that
√(x^2+a^2) + √(b^2+d^2) = 0
since even powers are always non-negative, that means that x,a,b,d are all zero.
and 0 is not in the interval (0,d)